Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Kunisch, Karl; Stadler, Georg

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 827-854.

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

Classification : 49M05, 49M29, 74M10, 74M15, 74B05
Mots clés : Signorini contact problems, Coulomb and Tresca friction, linear elasticity, semi-smooth Newton method, Fenchel dual, augmented lagrangians, complementarity system, active sets

[1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl. Mech. Engrg. 92 (1991) 353-375. | Zbl

[2] J. Alberty, C. Carstensen, S.A. Funken and R. Klose, Matlab implementation of the finite element method in elasticity. Computing 69 (2002) 239-263.

[3] A. Bensoussan and J. Frehse, Regularity results for nonlinear elliptic systems and applications, Springer-Verlag, Berlin. Appl. Math. Sci. 151 (2002). | MR | Zbl

[4] M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11 (2000) 495-521. | Zbl

[5] X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38 (2000) 1200-1216. | Zbl

[6] P.W. Christensen and J.S. Pang, Frictional contact algorithms based on semismooth Newton methods, in Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods, Kluwer Acad. Publ., Dordrecht. Appl. Optim. 22 (1999) 81-116. | Zbl

[7] P.W. Christensen, A. Klarbring, J.S. Pang and N. Strömberg, Formulation and comparison of algorithms for frictional contact problems. Internat. J. Numer. Methods Engrg. 42 (1998) 145-173. | Zbl

[8] Z. Dostál, J. Haslinger and R. Kučera, Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique. J. Comput. Appl. Math. 140 (2002) 245-256. | Zbl

[9] C. Eck and J. Jarušek, Existence results for the static contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 8 (1998) 445-468. | Zbl

[10] I. Ekeland and R. Témam, Convex analysis and variational problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Classics in Applied Mathematics 28 (1999). | MR | Zbl

[11] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, second edition. Grundlehren der Mathematischen Wissenschaften 224 (1983). | MR | Zbl

[12] R. Glowinski, Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. Springer-Verlag, New York (1984). | MR | Zbl

[13] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman (Advanced Publishing Program), Boston, MA. Monographs Stud. Math. 24 (1985). | MR | Zbl

[14] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, American Mathematical Society, Providence, RI. AMS/IP Studies in Advanced Mathematics 30 (2002). | MR | Zbl

[15] J. Haslinger, Approximation of the Signorini problem with friction, obeying the Coulomb law. Math. Methods Appl. Sci. 5 (1983) 422-437. | Zbl

[16] J. Haslinger, Z. Dostál and R. Kučera, On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2261-2281. | Zbl

[17] M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods. Math. Prog., Ser. B 101 (2004) 151-184. | Zbl

[18] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | Zbl

[19] M. Hintermüller, V. Kovtunenko and K. Kunisch, Semismooth Newton methods for a class of unilaterally constrained variational inequalities. Adv. Math. Sci. Appl. 14 (2004) 513-535. | Zbl

[20] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics. Springer, New York. Appl. Math. Sci. 66 (1988). | MR | Zbl

[21] S. Hüeber and B. Wohlmuth, A primal-dual active strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3147-3166. | Zbl

[22] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591-616. | Zbl

[23] K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41-62. | Numdam | Zbl

[24] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. SIAM Stud. Appl. Math. 8 (1988). | MR | Zbl

[25] A. Klarbring, Mathematical programming and augmented Lagrangian methods for frictional contact problems, A. Curnier, Ed. Proc. Contact Mechanics Int. Symp. (1992).

[26] R. Krause, Monotone Multigrid Methods for Signorini's Problem with Friction. Ph.D. Thesis, FU Berlin (2001).

[27] P. Laborde and Y. Renard, Fixed point strategies for elastostatic frictional contact problems. Rapport Interne 03-27, MIP Laboratory, Université Paul Sabatier, Toulouse (2003).

[28] A.Y.T. Leung, Guoqing Chen and Wanji Chen, Smoothing Newton method for solving two- and three-dimensional frictional contact problems. Internat. J. Numer. Methods Engrg. 41 (1998) 1001-1027. | Zbl

[29] C. Licht, E. Pratt and M. Raous, Remarks on a numerical method for unilateral contact including friction, in Unilateral problems in structural analysis, IV (Capri, 1989), Birkhäuser, Basel. Internat. Ser. Numer. Math. 101 (1991) 129-144. | Zbl

[30] J. Nečas, J. Jarušek and J. Haslinger, On the solution of the variational inequality to the Signorini problem with small friction. Boll. Unione Math. Ital. 5 (1980) 796-811. | Zbl

[31] C.A. Radoslovescu and M. Cocu, Internal approximation of quasi-variational inequalities. Numer. Math. 59 (1991) 385-398. | Zbl

[32] M. Raous, Quasistatic Signorini problem with Coulomb friction and coupling to adhesion, in New developments in contact problems, P. Wriggers and Panagiotopoulos, Eds., Springer Verlag. CISM Courses and Lectures 384 (1999) 101-178. | Zbl

[33] G. Stadler, Infinite-Dimensional Semi-Smooth Newton and Augmented Lagrangian Methods for Friction and Contact Problems in Elasticity. Ph.D. Thesis, University of Graz (2004).

[34] G. Stadler, Semismooth Newton and augmented Lagrangian methods for a simplified friction problem. SIAM J. Optim. 15 (2004) 39-62. | Zbl

[35] M. Ulbrich, Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2003) 805-842. | Zbl